Computational geometry: an introduction
Computational geometry: an introduction
How to net a lot with little: small &egr;-nets for disks and halfspaces
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Almost tight bounds for &egr;-nets
Discrete & Computational Geometry
Discrete & Computational Geometry - Special issue on ACM symposium on computational geometry, North Conway
Approximations and optimal geometric divide-and-conquer
Selected papers of the 23rd annual ACM symposium on Theory of computing
Fast stabbing of boxes in high dimensions
Theoretical Computer Science
Introduction to Algorithms
Lectures on Discrete Geometry
Output-sensitive construction of the union of triangles
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Improved approximation algorithms for geometric set cover
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
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Let ${\mathcal B}$ be any set of n axis-aligned boxes in ${\mathbb R}^{d}$, d≥ 1. We call a subset ${\mathcal N} \subseteq {\mathcal B}$ a (1/c)-net for ${\mathcal B}$ if any p ∈ ${\mathbb R}^{d}$ contained in more than n/c boxes of ${\mathcal B}$ must be contained in a box of ${\mathcal N}$, or equivalently if a point not contained in any box in ${\mathcal N}$ can only stab at most n/c boxes of ${\mathcal B}$. General VC-dimension theory guarantees the existence of (1/c)-nets of size O(clog c) for any fixed d, the constant in the big-Oh depending on d, and Matoušek [8, 9] showed how to compute such a net in time O(ncO(1)), or even O(n log c + cO(1)) which is O(n log c) if c is small enough. In this paper, we conjecture that axis-aligned boxes in ${\mathbb R}^{2}$ admit (1/c)-nets of size O(c), and that we can even compute such a net in time O(n log c), for any c between 1 and n. We show this to be true for intervals on the real line, and for various special cases (quadrants and skylines, which are unbounded in two and one directions respectively). In a follow-up version, we also show this to be true with various fatness We also investigate generalizations to higher dimensions.